Chaos in $SU(2)$ Yang-Mills Chern-Simons Matrix Model

We study the effects of addition of Chern-Simons (CS) term in the minimal Yang Mills (YM) matrix model composed of two $2 \times 2$ matrices with $SU(2)$ gauge and $SO(2)$ global symmetry. We obtain the Hamiltonian of this system in appropriate coordinates and demonstrate that its dynamics is sensitive to the values of both the CS coupling, $\kappa$, and the conserved conjugate momentum, $p_\phi$, associated to the $SO(2)$ symmetry. We examine the behavior of the emerging chaotic dynamics by computing the Lyapunov exponents and plotting the Poincar\'{e} sections as these two parameters are varied and, in particular, find that the largest Lyapunov exponents evaluated within a range of values of $\kappa$ are above that is computed at $\kappa=0$, for $\kappa p_\phi < 0$. We also give estimates of the critical exponents for the Lyapunov exponent as the system transits from the chatoic to non-chaotic phase with $p_\phi$ approaching to a critical value.